1 TheFastTrackIntroductiontoCalculusOscar E. Fernandez Subject Mathematics Keywords Calculus Simplified, Oscar E. Fernandez, Math, Mathematics, Princeton University Press …

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1 The Fast Track Introduction to Calculus

Chapter Preview. Calculus is a new way of thinking about mathematics. Thischapter provides you with a working understanding of the calculus mindset, core con-cepts of calculus, and the sorts of problems they help solve. The focus throughout is onthe ideas behind calculus (the big picture of calculus); the subsequent chapters discussthemath of calculus. After reading this chapter, you will have an intuitive understand-ing of calculus that will ground your subsequent studies of the subject. Ready? Let’s startthe adventure!

1.1 What Is Calculus?

Here’s my two-part answer to that question:

Calculus is a mindset—a dynamics mindset. Contentwise, calculus is themathematics of infinitesimal change.

Calculus as a Way of Thinking

The mathematics that precedes calculus—often called “pre-calculus,” which in-cludes algebra and geometry—largely focuses on static problems: problems devoidof change. By contrast, change is central to calculus—calculus is about dynamics.Example:

. What’s the perimeter of a square of side length 2 feet? ←− Pre-calculusproblem.

. How fast is the square’s perimeter changing if its side length is increasing atthe constant rate of 2 feet per second?←− Calculus problem.

This statics versus dynamics distinction between pre-calculus and calculus runseven deeper—change is themindset of calculus. The subject trains you to think of aproblem in terms of dynamics (versus statics). Example:

. Find the volume of a sphere of radius r. Pre-calculus mindset: Use 43πr

3

(Figure 1.1(a)).

. Find the volume of a sphere of radius r. Calculus mindset: Slice the sphereinto a gazillion disks of tiny thickness and then add up their volumes(Figure 1.1(b)). When the disks’ thickness is made “infinitesimally small” thisapproach reproduces the 4

3πr3 formula. (We will discuss why in Chapter 5.)

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2 • Introduction to Calculus

(a) (b)

Figure 1.1: Visualizing the volume of a sphere via (a) a pre-calculus mindset and (b) a calculusmindset.

There’s that mysterious word again—infinitesimal—and I’ve just given you a clueofwhat itmightmean. I’ll soon explain. Right now, letme pause to address a thoughtyou might have just had: “Why the slice-and-dice approach? Why not just use the43πr

3 formula?” The answer: had I asked for the volume of some random blob inspace instead, that static pre-calculus mindset wouldn’t have cut it (there is no for-mula for the volume of a blob). The dynamics mindset of calculus, on the otherhand, would have at least led us to a reasonable approximation using the sameslice-and-dice approach.

InteractiveF

igure

That volume example illustrates the power of the dynamics mindset of calculus.It also illustrates a psychological fact: shedding the static mindset of pre-calculus willtake some time. That was the dominant mindset in your mathematics courses priorto calculus mathematics courses, so you’re accustomed to thinking that way aboutmath. But fear not, young padawan (a Star Wars reference), I am here to guide youthrough the transition into calculus’ dynamicsmindset. Let’s continue the adventureby returning to what I’ve been promising: insight into infinitesimals.

What Does “Infinitesimal Change” Mean?

The volume example earlier clued you in to what “infinitesimal” might mean. Here’sa rough definition:

“Infinitesimal change” means: as close to zero change as you can imagine,but not zero change.

Let me illustrate this by way of Zeno of Elea (c. 490–430 BC), a Greek philosopherwho devised a set of paradoxes arguing that motion is not possible. (Clearly, Zenodid not have a dynamics mindset.) One such paradox—the Dichotomy Paradox—can be stated as follows:

To travel a certain distance you must first traverse half of it.

Figure 1.2 illustrates this. Here Zeno is trying to walk a distance of 2 feet. Butbecause of Zeno’s mindset, with his first step he walks only half the distance: 1 foot



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1.2 The Foundation of Calculus • 3

(a) (b) (c) (d)

0 2x

d = 1

∆d = 1

xd = 1.5

∆d = 0.5

xd = 1.75

∆d = 0.25

x

Figure 1.2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance witheach step.

(Figure 1.2(b)). He then walks half of the remaining distance in his second step: 0.5foot (Figure 1.2(c)). Table 1.1 keeps track of the total distance d, and the change indistance �d, after each of Zeno’s steps.

Table 1.1: The distance d and change in distance �d after each of Zeno’s steps.

�d 1 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125 · · ·d 1 1.5 1.75 1.875 1.9375 1.96875 1.984375 1.9921875 · · ·

Each change�d in Zeno’s distance is half the previous one. So as Zeno continues hiswalk,�d gets closer to zero but never becomes zero (because each�d is always halfof a positive number). If we checked back in with Zeno after he’s taken an infiniteamount of steps, the change �d resulting from his next step would be . . . drum rollplease . . . an infinitesimal change—as close to zero as you can imagine but not equalto zero.

This example, in addition to illustrating what an infinitesimal change is, also doestwomore things. First, it illustrates the dynamics mindset of calculus. We discussedZenowalking; we thought about the change in the distance he traveled; we visualizedthe situation with a figure and a table that each conveyedmovement. (Calculus is fullof action verbs!) Second, the example challenges us. Clearly, one can walk 2 feet.But as Table 1.1 suggests, that doesn’t happen during Zeno’s walk—he approachesthe 2-foot mark with each step yet never arrives. How do we describe this fact withan equation? (That’s the challenge.) No pre-calculus equation will do. We need anew concept that quantifies our very dynamic conclusion. That new concept is themathematical foundation of calculus: limits.

1.2 Limits: The Foundation of Calculus

Let’s return to Table 1.1. One thing you may have already noticed: �d and d arerelated. Specifically:

�d+ d= 2, or equivalently, d= 2−�d. (1.1)



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This equation relates each�d value to its corresponding d value in Table 1.1. Great.But it is not the equation we are looking for, because it doesn’t encode the dynamicsinherent in the table. The table clearly shows that the distance d traveled by Zenoapproaches 2 as �d approaches zero. We can shorten this to

d→ 2 as �d→ 0.

(We are using “→” here as a stand-in for “approaches.”) The table also reiterateswhatwe already know: were we to let Zeno continue his walk forever, he would be closerto the 2-foot mark than anyone could measure; in calculus we say: “infinitesimallyclose to 2.” To express this notion, we write

lim�d→0

d= 2, (1.2)

read “the limit of d as �d→ 0 (but is never equal to zero) is 2.”Equation (1.2) is the equation we’ve been looking for. It expresses the intuitive

idea that the 2-foot mark is the limiting value of the distance d Zeno’s traversing.(This explains the “lim” notation in (1.2).) Equation (1.2), therefore, is a statementabout the dynamics of Zeno’s walk, in contrast to (1.1), which is a statement aboutthe static snapshots of each step Zeno takes. Moreover, the Equation (1.2) remindsus that d is always approaching 2 yet never arrives at 2. The same idea holds for �d:it is always approaching 0 yet never arrives at 0. Said more succinctly:

Limits approach indefinitely (and thus never arrive).

Finite change ∆x in Y(Not a calculus concept)

Infinitesimal change in Y(Calculus concept)

lim∆x→0

Y

Figure 1.3: The calculus workflow.

We will learn much moreabout limits in Chapter 2 (in-cluding that (1.2) is actuallya “right-hand limit”). But theZeno example is sufficient togive you a sense of what thecalculus concept of limit is andhow it arises. It also illustratesthis section’s title—the limit concept is the foundation on which the entire mansionof calculus is built. Figure 1.3 illustrates the process of building a new calculus con-cept that we will use over and over again throughout the book: start with a finitechange �x in a quantity Y that depends on x, then shrink �x to zero without lettingit equal zero (i.e., take the limit as �x→ 0) to arrive at a calculus result. Workingthrough this process—like we just did with the Zeno example, and like you can nowrecognize in Figure 1.1—is part of what doing calculus is all about. This is what Imeant earlier when I said that calculus is the mathematics of infinitesimal change—contentwise, calculus is the collection of what results when we apply the workflowin Figure 1.3 to various quantities Y of real-world and mathematical interest.



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1.3 The Three Difficult Problems • 5

Three such quantities drove the historical development of calculus: instantaneousspeed, the slope of the tangent line, and the area under a curve. In the next sectionwe’ll preview how the calculus workflow in Figure 1.3 solved all of these problems.(We’ll fill in the details in Chapters 3–5.)

1.3 The Three Difficult Problems That Led to the Inventionof Calculus

Calculus developed out of a need to solve three Big Problems (refer to Figure 1.4):1

1. The instantaneous speed problem: Calculate the speed of a falling object at aparticular instant during its fall. (See Figure 1.4(a).)

2. The tangent line problem:Given a curve and a point P on it, calculate the slopeof the line “tangent” to the curve at P. (See Figure 1.4(b).)

3. The area under the curve problem: Calculate the area under the graph of afunction and bounded by two x-values. (See Figure 1.4(c).)

Figure 1.4 already gives you a sense of why these problems were so difficult tosolve—the standard approach suggested by the problem itself just doesn’t work. Forexample, you’ve been taught you need two points to calculate the slope of a line.The tangent line problem asks you to calculate the slope of a line using just onepoint (point P in Figure 1.4(b)). Similarly, we think of speed as “change in distancedivided by change in time.” How, then, can one possibly calculate the speed at aninstant, for which there is no change in time? These are examples of the sorts ofroadblocks that stood in the way of solving the three Big Problems.

(a) (b)

x x

yy

P

y = f (x)y = f (x)

a b

(c)

Speed at this instant = ? Slope of this line = ? Area of the shaded region = ?

Figure 1.4: The three problems that drove the development of calculus.

1These may not seem like important problems. But their resolution revolutionized science, enabling theunderstanding of phenomena as diverse as gravity, the spread of infectious diseases, and the dynamics of the worldeconomy.



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x

x

y

y

a

P

y = f (x)

y = f (x)

a b

Area is swept out …

x

yy = f (x)

a b

{

… up to ∆x past b …

∆x → 0

∆x

∆x → 0

∆t → 0

… with resulting areadenoted A∆x

x

yy = f (x)

a bInfinitesimally small ∆x

Infinitesimally small ∆x

Infinitesimally small ∆t

x

y

a

P

y = f (x)

Smaller ∆x

∆x∆y ∆y

∆y—∆x

∆xx

y

a

P

y = f (x)

Large ∆x

∆d

∆d∆d

Shaded areaunder the

curve:

Large ∆t Smaller ∆t Even smaller ∆t

lim∆x→0

A∆x

Slope of thetangent line:

lim∆x→0

∆d—∆t

Instantaneousspeed:

CalculusresultCalculus workflow

Limitingpicture

lim∆t→0

Figure 1.5: The calculus workflow (from Figure 1.3) applied to the three Big Problems.

But recall my first characterization of calculus: calculus is a dynamics mindset.Nothing about Figure 1.4 says “dynamics.” Every image is a static snapshot of some-thing (e.g., an area). So let’s calculus the figure. (Yep, I’m encouraging you to thinkof calculus as a verb.)

Figure 1.5 illustrates the application of the calculus workflow (from Figure 1.3)to each Big Problem. Each row uses a dynamics mindset to recast the problem as



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1.3 The Three Difficult Problems • 7

the limit of a sequence of similar quantities (e.g., slopes) involving finite changes.Specifically:

. Row #1: The instantaneous speed of the falling apple is realized as the limitof its average speeds �d

�t (ratios of changes in distance to changes in time) as�t→ 0.

. Row #2: The slope of the tangent line is realized as the limit of the slopes ofthe secant lines �y

�x (the gray lines in the figure) as �x→ 0.

. Row #3: The area under the curve is realized as the limit as �x→ 0 of thearea swept out from x= a up to �x past b.

The limit obtained in the second row of the figure is called the derivative of f (x)at x= a, the x-value of point P. The limit obtained in the third row of the figure iscalled the definite integral of f (x) between x= a and x= b. Derivatives and inte-grals round out the three most important concepts in calculus (limits are the third).We will discuss derivatives in Chapters 3 and 4 and integrals in Chapter 5, wherewe’ll also fill in themathematical details associatedwith the three limits in Figure 1.5.

This completes my big picture overview of calculus. Looking back now at Fig-ures 1.1, 1.2, and 1.5, I hope I’ve convinced you of the power of the calculus mindsetand the calculus workflow. We will employ both throughout the book. And becausethe notion of a limit is at the core of the workflow, I’ll spend the next chapter teach-ing you all about limits—their precise definition, the various types of limits, and themyriad techniques to calculate them. See you in the next chapter.



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1 TheFastTrackIntroductiontoCalculusOscar E. Fernandez Subject Mathematics Keywords Calculus Simplified, Oscar E. Fernandez, Math, Mathematics, Princeton University Press …